Abstract

In this Letter, a novel Fourier transform-based iterative method is proposed to reconstruct a refractive index distribution in an object directly from measured differential phase shifts. This reconstruction method offers distinct advantages over the analytic method in cases of noisy and/or few-view data, since the iterative scheme can easily incorporate compressive sensing and maximum likelihood techniques to achieve high image quality. Numerical simulation and biological sample experiments demonstrate the merits of the proposed approach.

© 2012 Optical Society of America

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  1. A. Momose, T. Takeda, Y. Itai, and K. Hirano, Nat. Med. 2, 473 (1996).
    [CrossRef]
  2. A. Momose, Jpn. J. Appl. Phys. 42, L866 (2003).
    [CrossRef]
  3. T. Weitkamp, A. Diaz, C. David, F. Pfeiffer, M. Stampanoni, P. Cloetens, and E. Ziegler, Opt. Express 13, 6296 (2005).
    [CrossRef]
  4. A. Momose, W. Yashiro, Y. Takeda, Y. Suzuki, and T. Hattori, Jpn. J. Appl. Phys. 45, 5254 (2006).
    [CrossRef]
  5. F. Pfeiffer, T. Weitkamp, O. Bunk, and C. David, Nat. Phys. 2, 258 (2006).
    [CrossRef]
  6. D. Chapman, W. Thomlinson, R. E. Johnston, D. Washburn, E. Pisano, N. Gmur, Z. Zhong, R. Menk, F. Arfelli, and D. Sayers, Phys. Med. Biol. 42, 2015 (1997).
    [CrossRef]
  7. K. M. Pavlov, C. M. Kewish, J. R. Davis, and M. J. Morga, J. Phys. D 34, A168 (2001).
    [CrossRef]
  8. A. Maksimenko, M. Ando, S. Hiroshi, and T. Yuasa, Appl. Phys. Lett. 86, 124105 (2005).
    [CrossRef]
  9. I. Koyama, A. Momose, J. Wu, T. Lwin, and T. Takeda, Jpn. J. Appl. Phys. 44, 8219 (2005).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  14. T. M. Buzug, Computed Tomography: From Photon Statistics to Modern Cone-Beam CT (Springer, 2008).
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    [CrossRef]
  16. D. M. Paganin, Coherent X-Ray Optics (Oxford University, 2006).

2011

T. Köhler, B. Brendel, and E. Roessl, Med. Phys. 38, 4542 (2011).
[CrossRef]

2008

G. Chen and Z. Qi, Phys. Med. Biol. 53, 1015 (2008).
[CrossRef]

F. Pfeiffer, C. David, O. Bunk, T. Donath, M. Bech, G. Le Duc, A. Bravin, and P. Cloetens, Phys. Rev. Lett. 101, 168101 (2008).
[CrossRef]

2006

Z. Huang, K. Kang, Z. Li, P. Zhu, Q. Yuan, W. Huang, J. Wang, D. Zhang, and A. Yu, Appl. Phys. Lett. 89, 041124 (2006).
[CrossRef]

A. Momose, W. Yashiro, Y. Takeda, Y. Suzuki, and T. Hattori, Jpn. J. Appl. Phys. 45, 5254 (2006).
[CrossRef]

F. Pfeiffer, T. Weitkamp, O. Bunk, and C. David, Nat. Phys. 2, 258 (2006).
[CrossRef]

2005

T. Weitkamp, A. Diaz, C. David, F. Pfeiffer, M. Stampanoni, P. Cloetens, and E. Ziegler, Opt. Express 13, 6296 (2005).
[CrossRef]

A. Maksimenko, M. Ando, S. Hiroshi, and T. Yuasa, Appl. Phys. Lett. 86, 124105 (2005).
[CrossRef]

I. Koyama, A. Momose, J. Wu, T. Lwin, and T. Takeda, Jpn. J. Appl. Phys. 44, 8219 (2005).
[CrossRef]

2003

A. Momose, Jpn. J. Appl. Phys. 42, L866 (2003).
[CrossRef]

2001

K. M. Pavlov, C. M. Kewish, J. R. Davis, and M. J. Morga, J. Phys. D 34, A168 (2001).
[CrossRef]

1997

D. Chapman, W. Thomlinson, R. E. Johnston, D. Washburn, E. Pisano, N. Gmur, Z. Zhong, R. Menk, F. Arfelli, and D. Sayers, Phys. Med. Biol. 42, 2015 (1997).
[CrossRef]

1996

A. Momose, T. Takeda, Y. Itai, and K. Hirano, Nat. Med. 2, 473 (1996).
[CrossRef]

1988

Ando, M.

A. Maksimenko, M. Ando, S. Hiroshi, and T. Yuasa, Appl. Phys. Lett. 86, 124105 (2005).
[CrossRef]

Arfelli, F.

D. Chapman, W. Thomlinson, R. E. Johnston, D. Washburn, E. Pisano, N. Gmur, Z. Zhong, R. Menk, F. Arfelli, and D. Sayers, Phys. Med. Biol. 42, 2015 (1997).
[CrossRef]

Bech, M.

F. Pfeiffer, C. David, O. Bunk, T. Donath, M. Bech, G. Le Duc, A. Bravin, and P. Cloetens, Phys. Rev. Lett. 101, 168101 (2008).
[CrossRef]

Bravin, A.

F. Pfeiffer, C. David, O. Bunk, T. Donath, M. Bech, G. Le Duc, A. Bravin, and P. Cloetens, Phys. Rev. Lett. 101, 168101 (2008).
[CrossRef]

Brendel, B.

T. Köhler, B. Brendel, and E. Roessl, Med. Phys. 38, 4542 (2011).
[CrossRef]

Bunk, O.

F. Pfeiffer, C. David, O. Bunk, T. Donath, M. Bech, G. Le Duc, A. Bravin, and P. Cloetens, Phys. Rev. Lett. 101, 168101 (2008).
[CrossRef]

F. Pfeiffer, T. Weitkamp, O. Bunk, and C. David, Nat. Phys. 2, 258 (2006).
[CrossRef]

Buzug, T. M.

T. M. Buzug, Computed Tomography: From Photon Statistics to Modern Cone-Beam CT (Springer, 2008).

Byer, R. L.

Chapman, D.

D. Chapman, W. Thomlinson, R. E. Johnston, D. Washburn, E. Pisano, N. Gmur, Z. Zhong, R. Menk, F. Arfelli, and D. Sayers, Phys. Med. Biol. 42, 2015 (1997).
[CrossRef]

Chen, G.

G. Chen and Z. Qi, Phys. Med. Biol. 53, 1015 (2008).
[CrossRef]

Cloetens, P.

F. Pfeiffer, C. David, O. Bunk, T. Donath, M. Bech, G. Le Duc, A. Bravin, and P. Cloetens, Phys. Rev. Lett. 101, 168101 (2008).
[CrossRef]

T. Weitkamp, A. Diaz, C. David, F. Pfeiffer, M. Stampanoni, P. Cloetens, and E. Ziegler, Opt. Express 13, 6296 (2005).
[CrossRef]

David, C.

F. Pfeiffer, C. David, O. Bunk, T. Donath, M. Bech, G. Le Duc, A. Bravin, and P. Cloetens, Phys. Rev. Lett. 101, 168101 (2008).
[CrossRef]

F. Pfeiffer, T. Weitkamp, O. Bunk, and C. David, Nat. Phys. 2, 258 (2006).
[CrossRef]

T. Weitkamp, A. Diaz, C. David, F. Pfeiffer, M. Stampanoni, P. Cloetens, and E. Ziegler, Opt. Express 13, 6296 (2005).
[CrossRef]

Davis, J. R.

K. M. Pavlov, C. M. Kewish, J. R. Davis, and M. J. Morga, J. Phys. D 34, A168 (2001).
[CrossRef]

Diaz, A.

Donath, T.

F. Pfeiffer, C. David, O. Bunk, T. Donath, M. Bech, G. Le Duc, A. Bravin, and P. Cloetens, Phys. Rev. Lett. 101, 168101 (2008).
[CrossRef]

Faris, G. W.

Gmur, N.

D. Chapman, W. Thomlinson, R. E. Johnston, D. Washburn, E. Pisano, N. Gmur, Z. Zhong, R. Menk, F. Arfelli, and D. Sayers, Phys. Med. Biol. 42, 2015 (1997).
[CrossRef]

Hattori, T.

A. Momose, W. Yashiro, Y. Takeda, Y. Suzuki, and T. Hattori, Jpn. J. Appl. Phys. 45, 5254 (2006).
[CrossRef]

Hirano, K.

A. Momose, T. Takeda, Y. Itai, and K. Hirano, Nat. Med. 2, 473 (1996).
[CrossRef]

Hiroshi, S.

A. Maksimenko, M. Ando, S. Hiroshi, and T. Yuasa, Appl. Phys. Lett. 86, 124105 (2005).
[CrossRef]

Huang, W.

Z. Huang, K. Kang, Z. Li, P. Zhu, Q. Yuan, W. Huang, J. Wang, D. Zhang, and A. Yu, Appl. Phys. Lett. 89, 041124 (2006).
[CrossRef]

Huang, Z.

Z. Huang, K. Kang, Z. Li, P. Zhu, Q. Yuan, W. Huang, J. Wang, D. Zhang, and A. Yu, Appl. Phys. Lett. 89, 041124 (2006).
[CrossRef]

Itai, Y.

A. Momose, T. Takeda, Y. Itai, and K. Hirano, Nat. Med. 2, 473 (1996).
[CrossRef]

Johnston, R. E.

D. Chapman, W. Thomlinson, R. E. Johnston, D. Washburn, E. Pisano, N. Gmur, Z. Zhong, R. Menk, F. Arfelli, and D. Sayers, Phys. Med. Biol. 42, 2015 (1997).
[CrossRef]

Kang, K.

Z. Huang, K. Kang, Z. Li, P. Zhu, Q. Yuan, W. Huang, J. Wang, D. Zhang, and A. Yu, Appl. Phys. Lett. 89, 041124 (2006).
[CrossRef]

Kewish, C. M.

K. M. Pavlov, C. M. Kewish, J. R. Davis, and M. J. Morga, J. Phys. D 34, A168 (2001).
[CrossRef]

Köhler, T.

T. Köhler, B. Brendel, and E. Roessl, Med. Phys. 38, 4542 (2011).
[CrossRef]

Koyama, I.

I. Koyama, A. Momose, J. Wu, T. Lwin, and T. Takeda, Jpn. J. Appl. Phys. 44, 8219 (2005).
[CrossRef]

Le Duc, G.

F. Pfeiffer, C. David, O. Bunk, T. Donath, M. Bech, G. Le Duc, A. Bravin, and P. Cloetens, Phys. Rev. Lett. 101, 168101 (2008).
[CrossRef]

Li, Z.

Z. Huang, K. Kang, Z. Li, P. Zhu, Q. Yuan, W. Huang, J. Wang, D. Zhang, and A. Yu, Appl. Phys. Lett. 89, 041124 (2006).
[CrossRef]

Lwin, T.

I. Koyama, A. Momose, J. Wu, T. Lwin, and T. Takeda, Jpn. J. Appl. Phys. 44, 8219 (2005).
[CrossRef]

Maksimenko, A.

A. Maksimenko, M. Ando, S. Hiroshi, and T. Yuasa, Appl. Phys. Lett. 86, 124105 (2005).
[CrossRef]

Menk, R.

D. Chapman, W. Thomlinson, R. E. Johnston, D. Washburn, E. Pisano, N. Gmur, Z. Zhong, R. Menk, F. Arfelli, and D. Sayers, Phys. Med. Biol. 42, 2015 (1997).
[CrossRef]

Momose, A.

A. Momose, W. Yashiro, Y. Takeda, Y. Suzuki, and T. Hattori, Jpn. J. Appl. Phys. 45, 5254 (2006).
[CrossRef]

I. Koyama, A. Momose, J. Wu, T. Lwin, and T. Takeda, Jpn. J. Appl. Phys. 44, 8219 (2005).
[CrossRef]

A. Momose, Jpn. J. Appl. Phys. 42, L866 (2003).
[CrossRef]

A. Momose, T. Takeda, Y. Itai, and K. Hirano, Nat. Med. 2, 473 (1996).
[CrossRef]

Morga, M. J.

K. M. Pavlov, C. M. Kewish, J. R. Davis, and M. J. Morga, J. Phys. D 34, A168 (2001).
[CrossRef]

Paganin, D. M.

D. M. Paganin, Coherent X-Ray Optics (Oxford University, 2006).

Pavlov, K. M.

K. M. Pavlov, C. M. Kewish, J. R. Davis, and M. J. Morga, J. Phys. D 34, A168 (2001).
[CrossRef]

Pfeiffer, F.

F. Pfeiffer, C. David, O. Bunk, T. Donath, M. Bech, G. Le Duc, A. Bravin, and P. Cloetens, Phys. Rev. Lett. 101, 168101 (2008).
[CrossRef]

F. Pfeiffer, T. Weitkamp, O. Bunk, and C. David, Nat. Phys. 2, 258 (2006).
[CrossRef]

T. Weitkamp, A. Diaz, C. David, F. Pfeiffer, M. Stampanoni, P. Cloetens, and E. Ziegler, Opt. Express 13, 6296 (2005).
[CrossRef]

Pisano, E.

D. Chapman, W. Thomlinson, R. E. Johnston, D. Washburn, E. Pisano, N. Gmur, Z. Zhong, R. Menk, F. Arfelli, and D. Sayers, Phys. Med. Biol. 42, 2015 (1997).
[CrossRef]

Qi, Z.

G. Chen and Z. Qi, Phys. Med. Biol. 53, 1015 (2008).
[CrossRef]

Roessl, E.

T. Köhler, B. Brendel, and E. Roessl, Med. Phys. 38, 4542 (2011).
[CrossRef]

Sayers, D.

D. Chapman, W. Thomlinson, R. E. Johnston, D. Washburn, E. Pisano, N. Gmur, Z. Zhong, R. Menk, F. Arfelli, and D. Sayers, Phys. Med. Biol. 42, 2015 (1997).
[CrossRef]

Stampanoni, M.

Suzuki, Y.

A. Momose, W. Yashiro, Y. Takeda, Y. Suzuki, and T. Hattori, Jpn. J. Appl. Phys. 45, 5254 (2006).
[CrossRef]

Takeda, T.

I. Koyama, A. Momose, J. Wu, T. Lwin, and T. Takeda, Jpn. J. Appl. Phys. 44, 8219 (2005).
[CrossRef]

A. Momose, T. Takeda, Y. Itai, and K. Hirano, Nat. Med. 2, 473 (1996).
[CrossRef]

Takeda, Y.

A. Momose, W. Yashiro, Y. Takeda, Y. Suzuki, and T. Hattori, Jpn. J. Appl. Phys. 45, 5254 (2006).
[CrossRef]

Thomlinson, W.

D. Chapman, W. Thomlinson, R. E. Johnston, D. Washburn, E. Pisano, N. Gmur, Z. Zhong, R. Menk, F. Arfelli, and D. Sayers, Phys. Med. Biol. 42, 2015 (1997).
[CrossRef]

Wang, J.

Z. Huang, K. Kang, Z. Li, P. Zhu, Q. Yuan, W. Huang, J. Wang, D. Zhang, and A. Yu, Appl. Phys. Lett. 89, 041124 (2006).
[CrossRef]

Washburn, D.

D. Chapman, W. Thomlinson, R. E. Johnston, D. Washburn, E. Pisano, N. Gmur, Z. Zhong, R. Menk, F. Arfelli, and D. Sayers, Phys. Med. Biol. 42, 2015 (1997).
[CrossRef]

Weitkamp, T.

Wu, J.

I. Koyama, A. Momose, J. Wu, T. Lwin, and T. Takeda, Jpn. J. Appl. Phys. 44, 8219 (2005).
[CrossRef]

Yashiro, W.

A. Momose, W. Yashiro, Y. Takeda, Y. Suzuki, and T. Hattori, Jpn. J. Appl. Phys. 45, 5254 (2006).
[CrossRef]

Yu, A.

Z. Huang, K. Kang, Z. Li, P. Zhu, Q. Yuan, W. Huang, J. Wang, D. Zhang, and A. Yu, Appl. Phys. Lett. 89, 041124 (2006).
[CrossRef]

Yuan, Q.

Z. Huang, K. Kang, Z. Li, P. Zhu, Q. Yuan, W. Huang, J. Wang, D. Zhang, and A. Yu, Appl. Phys. Lett. 89, 041124 (2006).
[CrossRef]

Yuasa, T.

A. Maksimenko, M. Ando, S. Hiroshi, and T. Yuasa, Appl. Phys. Lett. 86, 124105 (2005).
[CrossRef]

Zhang, D.

Z. Huang, K. Kang, Z. Li, P. Zhu, Q. Yuan, W. Huang, J. Wang, D. Zhang, and A. Yu, Appl. Phys. Lett. 89, 041124 (2006).
[CrossRef]

Zhong, Z.

D. Chapman, W. Thomlinson, R. E. Johnston, D. Washburn, E. Pisano, N. Gmur, Z. Zhong, R. Menk, F. Arfelli, and D. Sayers, Phys. Med. Biol. 42, 2015 (1997).
[CrossRef]

Zhu, P.

Z. Huang, K. Kang, Z. Li, P. Zhu, Q. Yuan, W. Huang, J. Wang, D. Zhang, and A. Yu, Appl. Phys. Lett. 89, 041124 (2006).
[CrossRef]

Ziegler, E.

Appl. Opt.

Appl. Phys. Lett.

Z. Huang, K. Kang, Z. Li, P. Zhu, Q. Yuan, W. Huang, J. Wang, D. Zhang, and A. Yu, Appl. Phys. Lett. 89, 041124 (2006).
[CrossRef]

A. Maksimenko, M. Ando, S. Hiroshi, and T. Yuasa, Appl. Phys. Lett. 86, 124105 (2005).
[CrossRef]

J. Phys. D

K. M. Pavlov, C. M. Kewish, J. R. Davis, and M. J. Morga, J. Phys. D 34, A168 (2001).
[CrossRef]

Jpn. J. Appl. Phys.

I. Koyama, A. Momose, J. Wu, T. Lwin, and T. Takeda, Jpn. J. Appl. Phys. 44, 8219 (2005).
[CrossRef]

A. Momose, Jpn. J. Appl. Phys. 42, L866 (2003).
[CrossRef]

A. Momose, W. Yashiro, Y. Takeda, Y. Suzuki, and T. Hattori, Jpn. J. Appl. Phys. 45, 5254 (2006).
[CrossRef]

Med. Phys.

T. Köhler, B. Brendel, and E. Roessl, Med. Phys. 38, 4542 (2011).
[CrossRef]

Nat. Med.

A. Momose, T. Takeda, Y. Itai, and K. Hirano, Nat. Med. 2, 473 (1996).
[CrossRef]

Nat. Phys.

F. Pfeiffer, T. Weitkamp, O. Bunk, and C. David, Nat. Phys. 2, 258 (2006).
[CrossRef]

Opt. Express

Phys. Med. Biol.

D. Chapman, W. Thomlinson, R. E. Johnston, D. Washburn, E. Pisano, N. Gmur, Z. Zhong, R. Menk, F. Arfelli, and D. Sayers, Phys. Med. Biol. 42, 2015 (1997).
[CrossRef]

G. Chen and Z. Qi, Phys. Med. Biol. 53, 1015 (2008).
[CrossRef]

Phys. Rev. Lett.

F. Pfeiffer, C. David, O. Bunk, T. Donath, M. Bech, G. Le Duc, A. Bravin, and P. Cloetens, Phys. Rev. Lett. 101, 168101 (2008).
[CrossRef]

Other

T. M. Buzug, Computed Tomography: From Photon Statistics to Modern Cone-Beam CT (Springer, 2008).

D. M. Paganin, Coherent X-Ray Optics (Oxford University, 2006).

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Figures (4)

Fig. 1.
Fig. 1.

Numerical phantom. (a) 2D numerical phantom and (b) the calculated sinogram of differential phase shifts of the phantom.

Fig. 2.
Fig. 2.

Reconstructions of numerical phantom. (a) The image reconstructed using the proposed Fourier transform-based iterative method and (b) the image reconstructed using analytic FBP method.

Fig. 3.
Fig. 3.

Comparison of profiles. Vertical line profiles through the center of the phantom (solid dark curve) and the reconstructed images with the proposed method (solid red curve) and FBP method (solid blue curve).

Fig. 4.
Fig. 4.

Reconstruction of biological sample. (a) The reconstructed image at the 200th slice using FBP from differential phase shifts of 251 projection views, (b) the image reconstructed using the proposed iterative method from differential phase shifts of 83 projection views, and (c) the image reconstructed using FBP method from differential phase shifts of 83 projection views. The display window is [0,1.1×107].

Equations (5)

Equations on this page are rendered with MathJax. Learn more.

Φ(s,θ)=2πλδ(sθ+tθ)dt,
Aθδ=bθ,
F(Aθ)δ=F(bθ),
F(sΦ(s,θ))=2πiwF(Φ)(w,θ).
(2πiwF(Aθ))δ=mθ,

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